Let $Γ$ be a finite graph and let $A(Γ)$ be the corresponding right-angled Artin group. From an arbitrary basis $\mathcal B$ of $H^1(A(Γ),\mathbb F)$ over an arbitrary field, we construct a natural graph $Γ_{\mathcal B}$ from the cup product, called the \emph{cohomology basis graph}. We show that $Γ_{\mathcal B}$ always contains $Γ$ as a subgraph. This provides an effective way to reconstruct the defining graph $Γ$ from the cohomology of $A(Γ)$, to characterize the planarity of the defining graph from the algebra of $A(Γ)$, and to recover many other natural graph-theoretic invariants. We also investigate the behavior of the cohomology basis graph under passage to elementary subminors, and show that it is not well-behaved under edge contraction.